↑

Properties

Label 2-546-273.62-c1-0-25
Degree $2$
Conductor $546$
Sign $0.252 + 0.967i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.0267 + 1.73i)3-s + (−0.499 + 0.866i)4-s − 0.655i·5-s + (1.48 − 0.889i)6-s + (0.402 − 2.61i)7-s + 0.999·8-s + (−2.99 + 0.0925i)9-s + (−0.567 + 0.327i)10-s + (−0.366 − 0.635i)11-s + (−1.51 − 0.842i)12-s + (0.424 − 3.58i)13-s + (−2.46 + 0.958i)14-s + (1.13 − 0.0175i)15-s + (−0.5 − 0.866i)16-s + (0.581 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0154 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.293i·5-s + (0.606 − 0.362i)6-s + (0.152 − 0.988i)7-s + 0.353·8-s + (−0.999 + 0.0308i)9-s + (−0.179 + 0.103i)10-s + (−0.110 − 0.191i)11-s + (−0.436 − 0.243i)12-s + (0.117 − 0.993i)13-s + (−0.659 + 0.256i)14-s + (0.293 − 0.00451i)15-s + (−0.125 − 0.216i)16-s + (0.141 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817526 - 0.631521i\)
\(L(\frac12)\) \(\approx\) \(0.817526 - 0.631521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.0267 - 1.73i)T \)
7 \( 1 + (-0.402 + 2.61i)T \)
13 \( 1 + (-0.424 + 3.58i)T \)
good5 \( 1 + 0.655iT - 5T^{2} \)
11 \( 1 + (0.366 + 0.635i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.581 + 1.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.51 + 4.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.55 - 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.839 + 0.484i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.20T + 31T^{2} \)
37 \( 1 + (-3.59 + 2.07i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.22 + 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.48iT - 47T^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + (2.15 + 1.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.932 - 0.538i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.07 - 2.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.09 - 7.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 + (-12.6 + 7.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.36 + 5.83i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49757031558600749744772791543, −9.950280535419670197761174102847, −9.037990219605428688608098694531, −8.180489879885448738652554047310, −7.28171391924030871955486144531, −5.71617313650280510466202414491, −4.72637623188270690893075153999, −3.80026272913165524947122544031, −2.77866586680246428714635292459, −0.71350182699492819988660399330, 1.57011737942573627363443752675, 2.80755067021464172494614172364, 4.59397936543820893129613354100, 5.96305412095130149232571124782, 6.32378765836490753712626232227, 7.48183604496891231745761951558, 8.189731392575325709995806957433, 8.971419460938638903468341190039, 9.907206105799582595179980931081, 11.08070348670630524582107615704

Graph of the $Z$-function along the critical line